Generalised dualities and maximal finite antichains in the homomorphism order of relational structures

The motivation for this paper is threefold. First, we study the connectivity properties of the homomorphism order of directed graphs, and more generally for relational structures. As opposed to the homomorphism order of undirected graphs (which has no non-trivial finite maximal antichains), the order of directed graphs has finite maximal antichains of any size. In this paper, we characterise explicitly all maximal antichains in the homomorphism order of directed graphs. Quite surprisingly, these maximal antichains correspond to generalised dualities. The notion of generalised duality is defined here in full generality as an extension of the notion of finitary duality, investigated in [J. Nešetřil, C. Tardif, Duality theorems for finite structures (characterising gaps and good characterisations), J. Combin. Theory Ser. B 80 (1) (2000) 80–97]. Building upon the results of the cited paper, we fully characterise the generalised dualities. It appears that these dualities are determined by forbidding homomorphisms from a finite set of forests (rather than trees). Finally, in the spirit of [A. Atserias, On digraph coloring problems and treewidth duality, in: Proceedings of the 21st IEEE Symposium on Logic in Computer Science, LICS’06, IEEE Computer Society, 2006; B. Larose, C. Loten, C. Tardif, A characterisation of first-order constraint satisfaction problems, in: Proceedings of the 21st IEEE Symposium on Logic in Computer Science, LICS’06, IEEE Computer Society, 2006; V. Dalmau, A. Krokhin, B. Larose, First-order definable retraction problems for posets and reflexive graphs, in: Proceedings of the 19th IEEE Symposium on Logic in Computer Science, LICS’04, IEEE Computer Society, 2004 [5]] we shall characterise “generalised” constraint satisfaction problems (defined also here) that are first-order definable. These are again just generalised dualities corresponding to finite maximal antichains in the homomorphism order

Dualities and dual pairs in Heyting algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction

Dismantlability, connectedness, and mixing in relational structures

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, in approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions and free energy. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Brice\~no. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.Comment: 27 pages, full version of the paper accepted to ICALP 201

Graph Relations and Constrained Homomorphism Partial Orders

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal